The Need for Computers
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With the vast range of mathematical tools available to the scientist, why are computers necessary? It is true that a scientist will likely try to represent the problem he is trying to solve using traditional (pencil and paper) mathematics. This approach often leads to solutions that are easy to understand - each step in the process unfolds as the calculations proceed. Then, the results of the calculations are often put into graphical form using computer techniques. The problem with this approach is that many physical problems simply do not lend themselves to hand (or, as we like to say, analytical) calculations. The difficulty may be that available mathematical techniques simply cannot do the job, or that the sheer quantity of calculations forbid meaningful results in reasonable times. Here's a simple example:

Solve the equation 3x + 7 = 10. Easy enough, x = 1. You can do it in your head - it took about 10 seconds.

Now solve the simultaneous equations 2x + 4y = 2; 3x - 2y = 1. Not bad, x = 1/2 and y = 1/4. It took about a minute.

Next, try the simultaneous equations 2x - 2y + 5z = 8   x - 3y + 2z = 4 3x + 7y + 5z = 8. The solutions are x = 9/5, y = - 1/5, z = 4/5 and, after I corrected some simple errors in addition, it took about 9 minutes to get the solutions. To the right is a plot of the number of variables vs. the time it took to solve for them. You can extrapolate to the case of more variables. Clearly, here's a case where a computer would come in handy. If the number of variables becomes large, computers become necessary. Do such large arrays of equations show up? Yes, indeed. In the calculation of nuclear properties, for example, the number of variables can easily be in the thousands. Notice that's not really hard to solve the equations, it's just time consuming (and boring). Computers never get bored. Also note another valuable use of computers - they allow you to draw nice pictures!

 Of course, the equations above are just those abstract problems you find in text books that don't relate to the real world. Here's some more practical problems - "word problems". Solve these, and keep track of two things: How long it took to get the word problem into a set of equations; How long it took to solve the equations.

1. Al's father is 45 years old. This is 15 more than twice Al's age.
How old is Al?
2. Frank is eight years older than his sister. In three years he will be
twice as old as she is. How old are they now?
3. Tom, Dick and Harry go to the casinos at Atlantic City. Their combined winnings were $14.00. Tom and Harry's winnings total 1 less than twice Dick's, while twice the sum of Tom's and Dick's winnings totals to one less than the combined winnings of Tom Dick and Harry. How much did each win?
4. In the 2003 NFL regular season, J. Lewis (Colts), A. Green (Packers) and D. McAllister totaled 5,209 yards rushing. Lewis and C. Portis (Broncos) totaled 3,543 yards. Green and McAllister totaled 3,257 yards while Portis and Green totaled 3256 yards. Rank the players in order of their individual rushing yards for the season, and find each player's total yards rushed.
5. Finally, here's a set of equations to solve - not a word problem. As a hint at the solution, if the values of each of the variables are multiplied by 1993 - the year STS-54 (Endeavour) was launched into orbit, they are all integers (5 positive, 1 negative). Otherwise, the equations have no significance whatsoever, except to convince you that computers can be an important time saver.

3a + 3b + 2c + d - 3e + 4f = -16,
a - 2b + 4c - 2d - e + 2f = -3,
4a - 3b - 2c - d + 5e - 9f = -2,
6a - 6b + 4 c - 7d + e + 6f = -6,
2a + 5b + 3c + 3d - 5e + 6f = -2,
4a - b - 6c + d - 4e - 2f = -5

Of course, these problems really aren't practical, but there are many applications that really need computers with high speeds. Presently speeds of a trillion operations per second are available, but there are applications that need thousands of times more speed: Real time computer imagery during surgery (NMR), computer simulation of new drug parameters, simulations in astrophysics and particle physics, modeling of environmental pollution patterns, long term climate changes and accurate weather prediction are just a few.